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(b) The second element in the specification of a force is its direction. The direction of a force is the line in which it acts. If the place of application of a force be regarded as a point, a line through that point, in the direction in which the force tends to move the body, is the direction of the force. In the case of a force distributed over a surface, it is frequently possible and convenient to assume a single point and a single line, such that a certain force acting at that point in that line would produce sensibly the same effect as is really produced.
Magnitude. (c) The third element in the specification of a force is its magnitude. This involves a consideration of the method followed in dynamics for measuring forces. Before measuring anything, it is necessary to have a unit of measurement, or a standard to which to refer, and a principle of numerical specification, or a mode of referring to the standard. These will be supplied presently. See also § 258, below.
Measure of force.
219. The Accelerative Effect of a Force is proportional to the velocity which it produces in a given time, and is measured by that which is, or would be, produced in unit of time; in other words, the rate of change of velocity which it produces. This is simply what we have already defined as acceleration, § 28.
220. The Measure of a Force is the quantity of motion which it produces per unit of time.
The reader, who has been accustomed to speak of a force of so many pounds, or so many tons, may be startled when he finds that such expressions are not definite unless it be specified at what part of the earth's surface the pound, or other definite quantity of matter named, is to be weighed; for the heaviness or gravity of a given quantity of matter differs in different latitudes. But the force required to produce a stated quantity of motion in a given time is perfectly definite, and independent of locality. Thus, let W be the mass of a body, g the velocity it would acquire in falling freely for a second, and P the force of gravity upon it, measured in kinetic or absolute units. We have
P = Wg.
221. According to the system commonly followed in mathe- Inconvenimatical treatises on dynamics till fourteen years ago, when a small of modern instalment of the first edition of the present work was issued for the use of our students, the unit of mass was g times the mass of the standard or unit weight. This definition, giving a varying and a very unnatural unit of mass, was exceedingly inconvenient. By taking the gravity of a constant mass for Standards the unit of force it makes the unit of force greater in high than are masses, in low latitudes. In reality, standards of weight are masses, primarily not forces. They are employed primarily in commerce for the measurepurpose of measuring out a definite quantity of matter; not an force. amount of matter which shall be attracted by the earth with a given force.
A merchant, with a balance and a set of standard weights, would give his customers the same quantity of the same kind of matter however the earth's attraction might vary, depending as he does upon weights for his measurement; another, using a spring-balance, would defraud his customers in high latitudes, and himself in low, if his instrument (which depends on constant forces and not on the gravity of constant masses) were correctly adjusted in London.
It is a secondary application of our standards of weight to employ them for the measurement of forces, such as steam pressures, muscular power, etc. In all cases where great accuracy is required, the results obtained by such a method have to be reduced to what they would have been if the measurements of force had been made by means of a perfect spring-balance, graduated so as to indicate the forces of gravity on the standard weights in some conventional locality.
It is therefore very much simpler and better to take the imperial pound, or other national or international standard weight, as, for instance, the gramme (see the chapter on Measures and Instruments), as the unit of mass, and to derive from it, according to Newton's definition above, the unit of force. This is the method which Gauss has adopted in his great improvement (§ 223 below) of the system of measurement of forces.
222. The formula, deduced by Clairault from observation, the amount and a certain theory regarding the figure and density of the
earth, may be employed to calculate the most probable value of the apparent force of gravity, being the resultant of true gravitation and centrifugal force, in any locality where no pendulum observation of sufficient accuracy has been made. This formula, with the two coefficients which it involves, corrected according to the best modern pendulum observations (Airy, Encyc. Metropolitana, Figure of the Earth), is as follows:
Let G be the apparent force of gravity on a unit mass at the equator, and g that in any latitude λ; then
g= G (1+005133 sin2 λ).
The value of G, in terms of the British absolute unit, to be explained immediately, is
According to this formula, therefore, polar gravity will be
223. Gravity having failed to furnish a definite standard, independent of locality, recourse must be had to something else. The principle of measurement indicated as above by Newton, but first introduced practically by Gauss, furnishes us with what we want. According to this principle, the unit force is that force which, acting on a national standard unit of matter during the unit of time, generates the unity of velocity.
This is known as Gauss's absolute unit; absolute, because it furnishes a standard force independent of the differing amounts of gravity at different localities. It is however terrestrial and inconstant if the unit of time depends on the earth's rotation, as it does in our present system of chronometry. The period of vibration of a piece of quartz crystal of specified shape and size and at a stated temperature (a tuning-fork, or bar, as one of the bars of glass used in the "musical glasses") gives us a unit of time which is constant through all space and all time, and independent of the earth. A unit of force founded on such a unit of time would be better entitled to the designation abso
lute than is the “absolute unit" now generally adopted, which is Maxwell's founded on the mean solar second. But this depends essentially tions for on one particular piece of matter, and is therefore liable to all Unit of the accidents, etc. which affect so-called National Standards however carefully they may be preserved, as well as to the almost insuperable practical difficulties which are experienced when we attempt to make exact copies of them. Still, in the present state of science, we are really confined to such approximations. The recent discoveries due to the Kinetic theory of gases and to Spectrum analysis (especially when it is applied to the light of the heavenly bodies) indicate to us natural standard pieces of matter such as atoms of hydrogen, or sodium, ready made in infinite numbers, all absolutely alike in every physical property. The time of vibration of a sodium particle corresponding to any one of its modes of vibration, is known to be absolutely independent of its position in the universe, and it will probably remain the same so long as the particle itself exists. The wavelength for that particular ray, i. e. the space through which light is propagated in vacuo during the time of one complete vibration of this period, gives a perfectly invariable unit of length; and it is possible that at some not very distant day the mass of such a sodium particle may be employed as a natural standard for the remaining fundamental unit. This, the latest improvement made upon our original suggestion of a Perennial Spring (First edition, § 406), is due to Clerk Maxwell*; who has also communicated to us another very important and interesting suggestion for founding the unit of time upon physical properties of a substance without the necessity of specifying any particular quantity of it. It is this, water being chosen as the substance of all others known to us which is most easily obtained in perfect purity and in perfectly definite physical condition.— Call the standard density of water the maximum density of the liquid when under the pressure of its own vapour alone. The time of revolution of an infinitesimal satellite close to the surface of a globe of water at standard density (or of any kind of matter at the same density) may be taken as the unit of time; for it is independent of the size of the globe. This has * Electricity and Magnetism, 1872.
Third sug. suggested to us still another unit, founded, however, still upon the same physical principle. The time of the gravest simple harmonic infinitesimal vibration of a globe of liquid, water at standard density, or of other perfect liquids at the same density, may be taken as the unit of time; for the time of the simple harmonic vibration of any one of the fundamental modes of a liquid sphere is independent of the size of the sphere.
Let f be the force of gravitational attraction between two units of matter at unit distance. The force of gravity at the surface of a globe of radius r, and density p, is fpr. Hence 3 if be the angular velocity of an infinitesimal satellite, we have, by the equilibrium of centrifugal force and gravity (SS 212, 477),
and therefore if T be the satellite's period,
(which is equal to the period of a simple pendulum whose length is the globe's radius, and weighted end infinitely near the surface of the globe). And it has been proved* that if a globe of liquid be distorted infinitesimally according to a spherical harmonic of order i, and left at rest, it will perform simple harmonic oscillations in a period equal to
2i+1) 4πƒp 2i (i-1))*
Hence if 7" denote the period of the gravest, that, namely, for which i = 2, we have
The semi-period of an infinitesimal satellite round the earth is equal, reckoned in seconds, to the square root of the number of metres in the earth's radius, the metre being very approximately
Dynamical Problems regarding Elastic Spheroidal Shells and Spheroids of Incompressible Liquid” (W. Thomson), Phil. Trans. Nov. 27, 1862.